\newproblem{lay:2_4_15}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 2.4.15}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
  When a deep space probe is launched, corrections may be necessary to place the probe on a precisely calculated trajectory. Radio telemetry
	provides a stream of vectors, $\mathbf{x}_1$, $\mathbf{x}_2$, ..., $\mathbf{x}_k$, giving information at different times about how the probe's
	position compares with its planned trajectory. Let $X_k$ be the matrix $\begin{pmatrix} \mathbf{x}_1 & \mathbf{x}_2 & ... & \mathbf{x}_k\end{pmatrix}$.
	The matrix $G_k=X_kX_k^T$ is computed as the radar data is analyzed. When $\mathbf{x}_{k+1}$ arrives a new $G_{k+1}$ must be computed. Since the
	data vectors arrive at high speed, the computational burden could be severe. But partitioned matrix multiplication helps tremendously.
	Compute the column-row expansions of $G_k$ and $G_{k+1}$, and describe what must be computed in order to \textit{update} $G_k$ to form $G_{k+1}$.
}{
  % Solution
	Let's analyze first $G_k$:
	\begin{center}
		$G_k=X_kX_k^T=\begin{pmatrix} \mathbf{x}_1 & \mathbf{x}_2 & ... & \mathbf{x}_k\end{pmatrix}
		              \begin{pmatrix} \mathbf{x}_1^T \\ \mathbf{x}_2^T \\ ... \\ \mathbf{x}_k^T\end{pmatrix} =
									\sum\limits_{i=1}^k{\mathbf{x}_i\mathbf{x}_i^T}$
	\end{center}
	Similarly
	\begin{center}
		$G_{k+1}=X_{k+1}X_{k+1}^T=\sum\limits_{i=1}^{k+1}{\mathbf{x}_i\mathbf{x}_i^T}=\left(\sum\limits_{i=1}^k{\mathbf{x}_i\mathbf{x}_i^T}\right)+
		   \mathbf{x}_{k+1}\mathbf{x}_{k+1}^T=G_k+\mathbf{x}_{k+1}\mathbf{x}_{k+1}^T$
	\end{center}
	Thus, it suffices to compute $\mathbf{x}_{k+1}\mathbf{x}_{k+1}^T$ and add it to the previous matrix $G_k$.
}
\useproblem{lay:2_4_15}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
